Geometric Sequences and
The Frequencies of the
88 Keys on a Piano
(The Equal Tempered
Chromatic Musical Scale)
By Don Cohen The Mathman
February,
2006
A
geometric sequence is found by starting with a number, then multiplying this by
a 'certain number'. Whatever one gets, you multiply that answer by that same
'certain number', and continue that. The
multiplying number is always the ratio of two consecutive numbers in the
sequence. For example, let's start with 3, and let our multiplying number
(that 'certain number' )or ratio, be 2, and write the first 4 terms of this
geometric sequence.
3,
3*2, 3*2*2,
3*2*2*2,.. , or using exponents: 3*2^{0}^{
},
3*2^{1}
, 3*2^{2}^{
},
3*2^{3}^{
}or
3, 6,12, 24, .. (Notice 2^{0}^{ }=1,
2^{1}^{
}=2, 2^{2}^{
}=4, 2^{3}^{
}=8).. What's the ratio of consecutive numbers in the
sequence? Well 6/3 = 2, 12/6 = 2,
and 24/12 = 2 ..
Problem 1: If the first number of a geometric sequence is 1 and the fourth number in the geometric sequence is 64, what are the second and third number in the geometric sequence, or what are the two geometric means between 1 and 64? 1, ? , ? , 64
Solution 1: 1= 1*r^{0}, (r being the ratio), the second number is 1*r^{1}, the third term is 1*r^{2} and the fourth number is 1*r^{3} = 64, so r^{3} = 64. What number cubed, or multiplied by itself 3 times = 64 or
^{what
is the cube root of 64 or}
^{?
4. Because 4*4*4 = 64, so r = 4. Our geometric sequence then is}
^{1, 4, 16,
64.}^{
}
Problem 2: If
the first number of a geometric sequence is 1 and the thirteenth number in the
geometric sequence is 2, what are the 2nd through 12^{th} numbers in the
geometric sequence, or the 11 geometric means between 1 and 2?
Solution 2:
1 2
3 4
5 6
7
8
9 10
11
12
13
1= r^{0}^{ },^{ }1*r^{1}^{ }, 1*r^{2}^{ }, 1*r^{3}^{ }, 1*r^{4}^{ }, 1*r^{5}^{ }, 1*r^{6}^{ }, 1*r^{7}^{ }, 1*r^{8}, 1*r^{9}^{ }, 1*r^{10}^{ }, 1*r^{11}^{ }, 1*r^{12}^{ }
^{
}1
, ?
, ?
,
? ,
?
,
?
, ?
,
? ,
?
,
?
, ? ,
?
,
2
^{So
r}^{12}^{
=
2
and r = }
^{= }^{1.05946'}
The subscript _{3} below indicates the 3^{rd} octave, if one starts with the 0^{th} octave A_{0}_{ }= 27.5 Hertz (vibrations/sec),_{ }
If
the first term, instead of 1, is middle C_{3} = 261.63 Hz, in the equal tempered
chromatic scale,
the second term is C_{3}^{#
}=
261.63*1.0594
= 261.63*
= 277.18 Hz and
D_{3}
= 261.63*1.0594^{2}^{
}=
261.63*
= 293.66 Hz and
D_{3}^{# }= 261.63*1.0594^{3}^{
}=
261.63*
= 311.12 Hz..
to
A_{4} = 261.63*1.0594^{9}^{
}=
261.63*
= 440.00 Hz ..
to
C_{4}^{
}= 261.63*1.0594^{12}
= 261.63*
= C_{ 3}*2
= 523.25
Hz. C_{4} is double the frequency of C_{ 3}_{ }, and an octave higher than C_{ 3}_{ .}
An octave is divided into
twelve intervals to form the equal tempered chromatic scale; the Intervals ^{of
pitch are described in terms of the ratios of the frequencies ~1.0594
=}
The frequency of each note
in the scale can be figured by multiplying each successive note by this number
to get the next. The frequency of any note can also be figured from A_{0}
= 27.5 Hz, using the formula:
f(N) = 27.5*2^{(N/12)},
where N is an index into the equal tempered chromatic scale notes
starting with N= 0 for A_{0},
the lowest note on the keyboard. N increases by 1 for each note on the keyboard. Note
that the actual key on the keyboard is N + 1. The 88^{th} key has a frequency for N=87 in the
formula above, so f(87) = 27.5*2^{(87/12)}
= 4186.009 Hz, the frequency of the note C_{7}
The table below
shows the note, the index number (N) and corresponding frequencies (f) and the Octave
for all of the keyboard notes based on A_{4}
= 440
Hz in the equal tempered chromatic scale.
Oct 

0 

1 

2 

3 

4 

5 

6 

7 
Note 
N 
f 
N 
f 
N 
f 
N 
f 
N 
f 
N 
f 
N 
f 
N 
f 
A 
0 
27.500 
12 
55.0000 
24 
110.0000 
36 
220.0000 
48 
440.0000 
60 
880.0000 
72 
1760.000 
84 
3520.000 
A# 
1 
29.135 
13 
58.2705 
25 
116.5409 
37 
233.0819 
49 
466.1638 
61 
932.3275 
73 
1864.655 
85 
3729.310 
B 
2 
30.867 
14 
61.7354 
26 
123.4708 
38 
246.9417 
50 
493.8833 
62 
987.7666 
74 
1975.533 
86 
3951.066 
C 
3 
32.703 
15 
65.4064 
27 
130.8128 
39 
261.6256 
51 
523.2511 
63 
1046.502 
75 
2093.005 
87 
4186.009 
C# 
4 
34.647 
16 
69.2957 
28 
138.5913 
40 
277.1826 
52 
554.3653 
64 
1108.731 
76 
2217.461 
88 
4434.922 
D 
5 
36.708 
17 
73.4162 
29 
146.8324 
41 
293.6648 
53 
587.3295 
65 
1174.659 
77 
2349.318 
89 
4698.636 
D# 
6 
38.890 
18 
77.7817 
30 
155.5635 
42 
311.1270 
54 
622.2540 
66 
1244.508 
78 
2489.016 
90 
4978.032 
E 
7 
41.203 
19 
82.4069 
31 
164.8138 
43 
329.6276 
55 
659.2551 
67 
1318.510 
79 
2637.020 
91 
5274.041 
F 
8 
43.653 
20 
87.3071 
32 
174.6141 
44 
349.2282 
56 
698.4565 
68 
1396.913 
80 
2793.826 
92 
5587.652 
F# 
9 
46.249 
21 
92.4986 
33 
184.9972 
45 
369.9944 
57 
739.9888 
69 
1479.978 
81 
2959.955 
93 
5919.911 
G 
10 
48.999 
22 
97.9989 
34 
195.9977 
46 
391.9954 
58 
783.9909 
70 
1567.982 
82 
3135.963 
94 
6271.927 
G# 
11 
51.913 
23 
103.8262 
35 
207.6523 
47 
415.3047 
59 
830.6094 
71 
1661.219 
83 
3322.438 
95 
6644.875 
The modern piano has 88 keys (seven octaves = 7*12 = 84, then add 4 to get
88. The 4 is a minor third = (1/3)*12= 4; so the 88 piano key notes go from from
A_{0},
the lowest note, or 27.5 Hz, to C_{7}_{,
}the highest note on the piano, 4186.009
Hz.
The following graph done in Mathematica, shows a polar graph of the frequencies for the octave from middle C _{ 3}_{ }(261.63) to C_{4}_{ }(523.25)._{ }Each point has as its distance from the origin (or radius) the pitch or frequency, and the angle from the starting position is (360/13)*n, starting with n=0, going counterclockwise. The command to do this graph is: PolarListPlot[Table[261.6256*2^(n/12),{n,0,12}],PlotStyle>PointSize[0.02]].
In trying to understand how all this works, Don found these two sources very
helpful:
'The Handbook of Chemistry and Physics' (26^{th} edition) edited
by Charles D. Hodgman and Harry N. Holmes, Published in 1942 by Chemical Rubber
Publishing Co., Cleveland, OH
An internet file 'The Equal Tempered Scale and Some Peculiarities of Piano Tuning' with the table above, by Jim Campbell